Question

find the exact value by using a half - angle identity. tan(75°) a. - 2 - \sqrt{3} b. 2 + \sqrt{3} c. 2 - \sqrt{3} d. - 2 + \sqrt{3}

Explanation:

Step1: Use the half - angle formula for tangent

The half - angle formula for tangent is \(\tan\frac{\alpha}{2}=\frac{1 - \cos\alpha}{\sin\alpha}\).
Since \(75^{\circ}=\frac{150^{\circ}}{2}\), then \(\tan75^{\circ}=\tan\frac{150^{\circ}}{2}\), and \(\cos150^{\circ}=-\frac{\sqrt{3}}{2}\), \(\sin150^{\circ}=\frac{1}{2}\).

Step2: Substitute the values into the formula

Substitute \(\cos150^{\circ}\) and \(\sin150^{\circ}\) into \(\tan\frac{\alpha}{2}=\frac{1 - \cos\alpha}{\sin\alpha}\), we get \(\tan75^{\circ}=\frac{1-(-\frac{\sqrt{3}}{2})}{\frac{1}{2}}\).
Simplify the numerator: \(1+\frac{\sqrt{3}}{2}=\frac{2 + \sqrt{3}}{2}\).
Then \(\tan75^{\circ}=\frac{\frac{2+\sqrt{3}}{2}}{\frac{1}{2}}=2+\sqrt{3}\).

Answer:

B. \(2+\sqrt{3}\)